A NEW VARIATIONAL PRINCIPLE AND ITS APPLICATION TO NONLINEAR HETEROGENEOUS SYSTEMS

A new variational principle is proposed for estimating the effective properties of nonlinear heterogeneous systems. The variational principle expresses the effective energy-density function of a given nonlinear heterogeneous dielectric in terms of an infinite-dimensional optimization problem involving the effective energy-density functions of the class of linear heterogeneous comparison materials with variable dielectric coefficients. The variational principle can alternatively be used in an approximate fashion-by optimizing over finite-dimensional subspaces of the original space of comparison dielectric coefficients-to obtain bounds on the effective dielectric properties of nonlinear composite materials with isotropic constituent phases in given proportions. The application of the procedure is illustrated by computing an exact estimate for the effective energy-density function of a nonlinear laminated dielectric material (which has overall anisotropic effective behavior). It is also demonstrated that the Talbot-Willis nonlinear extension of the Hashin-Shtrikman variational principle can be generated from the new variational principle by estimating the effective energy of the linear comparison material (in the new variational principle) directly from the Hashin-Shtrikman variational principle for linear materials. However, the new variational principle is more general in the sense that it can be used in conjunction with other bounds and estimates for the linear comparison material to yield corresponding bounds and estimates for nonlinear composites.